arXiv research

A new metric-based principal curve method learns 1D manifolds from spatial data.

PFs and iPFs learn principal manifolds for efficient density estimation.

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $α$-structure of frame bundles over statistical manifolds with respect to $α$-connections, by giving geometric structures. The man…

New Q-manifolds theory integrates Lie algebroids.

The paper shows nearly-Fuchsian properties for certain hyperbolic 3-manifolds.

We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic me…

Estimates for $p$-capacities on symmetric manifolds.

New approach for principal curves on spherical data.

New bounds found for minimal surfaces in hyperbolic 3-manifolds.

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…

R-PCA extends PCA to Riemannian manifolds for structured data.

Study characterizes martingales on fiber bundles for harmonic map analysis.

We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minim…

Study shows conditions for rational ellipticity of manifolds with symmetries.

In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found 'lines and planes of closest fit to system of points'. The famous k…

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given i…

Given a complex manifold $M$ equipped with a holomorphic action of a connected complex Lie group $G$, and a holomorphic principal $H$--bundle $E_H$ over $X$ equipped with a $G$--connection $h$, we investigate the connections on the principal $H$--bundle $E_H$ that are (strongly) adapted to $h$. Examples are provided by…

We construct a new example of an A-manifold, i.e. a Riemannian manifold with a cyclic-parallel Ricci tensor, which can be viewed as a generalization of the Einstein condition. The underlying manifold for our construction is a principal torus bundle over Kähler-Einstein manifold a with fibre a torus of arbitrary dimensi…

Introduces Lie semiheaps and their relation to Lie groups and bundles.

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that the Lie group action must have odd cohomogeneity. We pay special attention to mani…

New SKT manifolds created using toric geometry.

Introduces principal bundles in a new geometric category.

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this paper, we study the differential geometry of these objects, including connections …

We classify SO(n)-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(Π, G)$-bundles over cohomogeneity one manifolds.

Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality…

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_G$. We define pseudo-real principal $G$--bundles on $X$; these are generalizations of re…

While the existence of low-dimensional embedding manifolds has been shown in patterns of collective motion, the current battery of nonlinear dimensionality reduction methods are not amenable to the analysis of such manifolds. This is mainly due to the necessary spectral decomposition step, which limits control over the…

A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ o…

Let $P$ be a parabolic subgroup of a connected simply connected complex semisimple Lie group $G$. Given a compact Kähler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduct…

Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and surfaces. However, the problem of unwrapping or unfolding manifolds has received r…

In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order…

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of examples to which these existence results apply.

Unified method to compute Laplace spectra on homogeneous principal bundles.

Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.

The main result we give in this brief note relates, under suitable hypotheses, the φ-null Osserman, the null Osserman and the classical Osserman conditions to each other, via semi-Riemannian submersions as projection maps of principal torus bundles arising from a Lorentzian S-manifold.

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there i…

We investigate principal $G$-bundles on a compact Kähler manifold, where $G$ is a complex algebraic group such that the connected component of it containing the identity element is reductive. Defining (semi)stability of such bundles, it is shown that a principal $G$-bundle $E_G$ admits an Einstein-Hermitian connection …

We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lag…

Study numerically flat bundles on Fujiki manifolds using algebraic groups.

New metrics found on non-Kähler Calabi-Yau manifolds.

We study geometric first order differential operators on quaternionic Kähler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\Sp(1)\Sp(n)$. This observation leads to anti-symmetry of the principal symbols and Bochner-Weitzenböck formulas for operators. As an applicatio…

Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.

In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…

Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compa…

The paper studies regular contact manifolds and their products.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.